On generalized semi-bent (and partially bent) Boolean functions

In this paper, we obtain a characterization of generalized Boolean functions based on spectral analysis. We investigate a relationship between the Walsh-Hadamard spectrum and σ f , the sum-of-squares-modulus indicator (SSMI) of the generalized Boolean function. It is demonstrated that σ f = 22n+s for every s-plateaued generalized Boolean function in n variables. Two classes of generalized semi-bent Boolean functions are constructed.We have constructed a class of generalized semi-bent functions in (n + 1) variables from generalized semi-bent functions in n variables and identify a subclass of it for which σ f and 4 f both have optimal value. Finally, some construction on generalized partially bent Boolean functions are given.